The fermionic operators
can be reconstructed from the half-vertex operators and the charge and
shift operators \(C\)  and \(R\)  (see e.g. [1]}) via the so-called
boson–fermion correspondence:
\( \begin{split}\psi (z) = z^{C-\frac{1}{2}} R\, \Gamma _{-}(z) \Gamma _{+}^{-1} \left(z^{-1} \right),\qquad \psi ^*(w) = R^{-1} w^{-C+\frac{1}{2}} \Gamma _{-}^{-1}(w) \Gamma _{+}\left( w^{-1} \right).\end{split}\) 
